Steve Ballmer’s announcement last week of his retirement from Microsoft Corp. after 13 years at the helm leaves some puzzling questions for his successor—about Microsoft’s epic decline, its uncertain future and the ultimate purpose of a public company. Resolving them may well prove impossible. And that’s OK.

Under Ballmer, who succeeded Bill Gates as chief executive officer in 2000, Microsoft failed to anticipate or adequately respond to almost every technological innovation that came along, from Web search to music to social networking to mobile phones to tablet computers. Its core products—the Windows operating system and the Office suite of applications—continued to rack up big profits. Yet bureaucratic complexity deepened, top talent defected, and the corporate culture resisted change.

By the time Ballmer engineered the company’s latest management shakeup this year, the damage had been done. Microsoft has delivered to investors a total return of minus 1.6% since Ballmer took over, compared with a gain of 2.8% for its peers. PC shipments are declining, Windows increasingly looks like a poor fit for the smartphone era, and Surface, the company’s late entrant into the tablet market, has been a dud.

All of which makes Microsoft look like the embodiment of Clayton Christensen’s innovator’s dilemma, in which profitable companies fight so hard to protect their established products that they fail to respond to innovations by smaller and nimbler competitors. To react more aggressively, Ballmer would’ve risked disrupting the parts of Microsoft that were still making a lot of money—and will be for some time. It’s hard to blame him for resisting. Yet he leaves his successor in an unenviable position.

Microsoft still has $77 billion in cash and short-term investments, global name recognition, more than a billion users and average sales of about $150,000 every minute. But sustaining the company in the long run will require an ambitious rethink. Following the lead of Apple Inc. and International Business Machines Corp. (IBM), which have managed to reinvent themselves spectacularly, will require Microsoft to evolve from a company focused on specific products and into one focused on an idea that can last—even as new technologies come and go. For Apple, the idea is elegant and expensive consumer electronics; for IBM, it’s technology services for business.

What could it be for Microsoft? Ballmer himself has talked up a transition into a devices and services company. That’s awfully vague as far as defining ideas go. But it may be a reflection of how disparate Microsoft’s products have become: In addition to Windows, Office and tablets, it makes enterprise products, Xbox video-game consoles, a cloud-computing platform called Azure, an unprofitable search engine called Bing and a menagerie of other poorly named products of variable quality and popularity.

In the short term, the company could refocus on the part of its business that’s growing healthily—enterprise products—and invest in big data and cloud computing. Sales of such products increased 9% in the second quarter even as Windows and PC sales declined.

Down the road, if no larger organizing purpose materializes—a distinct possibility—Microsoft should consider breaking itself up. There’s no shortage of suggestions about how to do so. Rick Sherlund, an analyst at Nomura Securities, for example, has argued that selling the Bing and Xbox units, with attendant streamlining and job cuts, would allow Microsoft to substantially boost its dividend to shareholders. Other analysts suggest splitting Microsoft into one company for business products and another for consumers. Smaller and more focused companies that could experiment creatively, without fear of threatening entrenched products, might well thrive.

There would be no dishonour in going down this road. On the contrary, it could be a model for aging companies—which, like people and products, have useful lifespans. Hot new products will come and go, irrational consumer fads will wax and wane, and companies will rise and fall in mysterious ways. Sometimes admitting defeat is the best way to ensure your ideas endure.